关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.

On $\mathcal{F}$-S-Supplemented Subgroups

Let $\mathcal{F}$ be a class of groups. A subgroup $H$ of a group $G$ is called $\mathcal{F}$-S-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $K/K\cap{H_G}\in\mathcal{F}$, where $H_G=\bigcap_{g\in G}H^g$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, By using $\mathcal{F}$-S-supplemented subgroups, we give some new criteria for the solvability, nilpotency and supersolvability of finite groups. By these results, we may get a series of corollaries, which contain some known results.